📚 Pre-U OCR Psychology: Formulas and Theorems Quick Reference Handbook | Pre-U OCR心理学：公式定理速查手册
This handbook collates the essential quantitative formulas, psychophysical laws, and statistical theorems that appear regularly in the Pre-U OCR Psychology specification. It is designed as a rapid revision tool, with each entry accompanied by brief explanations and practical contexts in which the formula is applied. Mastering these tools will strengthen your ability to evaluate research evidence and design rigorous investigations.
本手册汇集了 Pre-U OCR 心理学课程中频繁出现的核心量化公式、心理物理定律和统计定理。它被设计为一本快速复习工具，每个条目都配有简要说明和公式应用的实际情境。熟练掌握这些工具将增强你评估研究证据和设计严谨研究的能力。
1. Weber’s Law | 韦伯定律
Weber’s Law states that the just noticeable difference (JND) between two stimuli is proportional to the magnitude of the standard stimulus. The constant of proportionality, known as the Weber fraction, varies across sensory modalities and reflects the resolution of a sensory system.
韦伯定律指出，两个刺激之间的最小可觉差（JND）与标准刺激的强度成正比。这个比例常数被称为韦伯分数，它在不同的感觉通道中取值不同，反映了一个感觉系统的分辨能力。
ΔI / I = k
where ΔI is the increment threshold (the smallest added intensity that can be detected), I is the intensity of the standard stimulus, and k is the Weber fraction. For example, for lifted weights, k ≈ 0.02, meaning a 100 g weight must be increased by about 2 g to be noticed as heavier.
其中 ΔI 为增量阈值（能被察觉的最小附加强度），I 为标准刺激的强度，k 为韦伯分数。例如，对于提重实验，k ≈ 0.02，这意味着一个 100 克的重量必须增加约 2 克才能被感知为更重。
The law holds well except at very low intensities, where the Weber fraction rises, indicating poorer discrimination near the absolute threshold. This deviation led to the development of Fechner’s logarithmic scaling.
该定律在强度很低时失效，韦伯分数会上升，表明在绝对阈限附近辨别能力较差。这一偏差促成了费希纳对数标度的提出。
2. Fechner’s Law | 费希纳定律
Building on Weber’s work, Gustav Fechner proposed that the magnitude of a sensation is proportional to the logarithm of the physical stimulus intensity. In other words, equal ratios of stimulus intensity produce equal increments in perceived sensation.
在韦伯工作的基础上，古斯塔夫·费希纳提出感觉强度与物理刺激强度的对数成正比。换言之，刺激强度的等比变化会产生感觉的等量增加。
S = k log I
where S is the sensation magnitude, I is the physical intensity, and k is a modality‑specific constant derived from the Weber fraction. Fechner assumed that all JNDs are subjectively equal units of sensation, integrating Weber’s law to obtain this logarithmic relationship.
其中 S 为感觉量，I 为物理强度，k 为由韦伯分数导出的模态特定常数。费希纳假设所有最小可觉差在主观上都是相等的感觉单位，通过积分韦伯定律得出这一对数关系。
Fechner’s law successfully accounts for brightness and loudness perception over moderate ranges, but fails at extremely high or low intensities. It also sparked a debate about whether sensation scales are logarithmic or power functions.
费希纳定律在中等的强度范围内很好地解释了亮度和响度知觉，但在极高或极低强度下失效。它也引发了感觉量表是对数函数还是幂函数的争论。
3. Stevens’ Power Law | 斯蒂文斯幂定律
S. S. Stevens, using direct magnitude estimation, argued that the relationship between physical intensity and perceived magnitude follows a power function. This law better captures the expansiveness of sensations like electric shock and the compressiveness of brightness.
S. S. 斯蒂文斯通过数量估计法提出，物理强度与知觉量之间的关系遵循幂函数。该定律能更好地捕捉电击等感觉的扩张性以及亮度的压缩性。
S = k Iⁿ
where the exponent n determines the shape of the psychophysical function. When n < 1, sensation grows more slowly than intensity (e.g., brightness, n ≈ 0.33); when n > 1, sensation accelerates (e.g., electric shock, n ≈ 3.5). The constant k scales the units.
其中指数 n 决定了心理物理函数的形状。当 n < 1 时，感觉的增长慢于物理强度的增长（如亮度，n ≈ 0.33）；当 n > 1 时，感觉加速增长（如电击，n ≈ 3.5）。常数 k 用于调节单位。
Stevens’ method of direct scaling avoided the assumption that JNDs are equal sensory units. His power law is now the standard in psychophysics, though it coexists with Fechner’s approach in specific contexts.
斯蒂文斯的直接量表法避免了“最小可觉差是相等感觉单位”的假设。尽管他的幂定律如今已成为心理物理学的标准，但在特定情境下仍与费希纳的方法并存。
4. Signal Detection Theory: Sensitivity and Bias | 信号检测论：敏感性与反应偏差
Signal detection theory (SDT) distinguishes between an observer’s sensory sensitivity and their decision criterion. It provides an objective measure of discriminability, d’ (d-prime), that is uncontaminated by response bias.
信号检测论（SDT）将观察者的感觉敏感度与其决策标准区分开来。它提供了一种不受反应偏差影响的客观辨别力测量指标——d’（d-prime）。
d’ = z(Hit) – z(False Alarm)
where z(Hit) and z(False Alarm) are the z‑scores corresponding to the hit rate and false alarm rate. Larger d’ values indicate greater sensitivity; a d’ of 0 signals chance performance. The response criterion, c, is often calculated as c = -0.5[z(Hit) + z(False Alarm)], with negative c indicating a liberal bias (more ‘yes’ responses).
其中 z(Hit) 和 z(False Alarm) 分别是击中率和虚惊率对应的 z 分数。d’ 值越大表示敏感性越高；d’ 为 0 表示随机表现。反应标准 c 常计算为 c = -0.5[z(Hit) + z(False Alarm)]，负值 c 表示宽松标准（更多“是”反应）。
SDT is widely used when analysing recognition memory experiments, eyewitness identification, and any scenario where a participant must discriminate signal from noise. It is a powerful alternative to simple percentage correct, as it teases apart the processes of discrimination and decision.
信号检测论广泛应用于分析再认记忆实验、目击者辨认以及任何需要被试从噪音中辨别信号的场景。它将辨别过程与决策过程分离开来，是简单正确率的有力替代。
5. Pearson’s Product-Moment Correlation | 皮尔逊积差相关
Pearson’s r measures the strength and direction of a linear relationship between two continuous variables. It is a parametric statistic that assumes interval‑level data, linearity, and homoscedasticity.
皮尔逊 r 测量两个连续变量之间线性关系的强度和方向。它是一种参数统计量，假设数据为等距水平、线性关系和方差同质性。
r = Σ(xy) / √(Σ x² × Σ y²)
where x = X – X̄ and y = Y – Ÿ, representing the deviations of each raw score from its respective mean. The coefficient ranges from -1 to +1, with 0 indicating no linear correlation. The squared value, r², gives the proportion of variance shared by the two variables.
其中 x = X – X̄, y = Y – Ÿ，表示每个原始分数与其各自均值的离差。系数取值范围从 -1 到 +1，0 表示没有线性相关。r² 表示两个变量共享的方差比例。
A significant r must be tested against the null hypothesis using a t‑test: t = r√[(n-2)/(1-r²)] with df = n-2. In Pre-U Psychology, Pearson’s r often appears when examining correlations between questionnaire scores or reaction times.
显著性 r 需要用 t 检验对零假设进行检验：t = r√[(n-2)/(1-r²)]，自由度 df = n-2。在 Pre-U 心理学中，皮尔逊 r 常用于考察问卷得分或反应时之间的相关。
6. Spearman’s Rank Correlation | 斯皮尔曼等级相关
Spearman’s rho (ρ or rₛ) is a non‑parametric measure of association for ordinal data or when the assumptions of Pearson’s r are violated. It assesses how well the relationship between two variables can be described by a monotonic function.
斯皮尔曼 ρ（或 rₛ）是一种用于顺序数据或当皮尔逊 r 的假设被违反时的非参数关联测量。它评估两个变量之间的关系在多大程度上可以被单调函数描述。
rₛ = 1 – (6 Σ D²) / [n(n² – 1)]
where D is the difference between the two ranks of each participant, and n is the number of pairs. Like Pearson’s r, rₛ ranges from -1 to +1. It is less sensitive to outliers and does not assume normality.
其中 D 为每个被试两个等级之差，n 为对数。与皮尔逊 r 一样，rₛ 的范围为 -1 到 +1。它对异常值不太敏感，且不假设正态性。
When many tied ranks exist, a correction factor is applied. Psychologists use Spearman’s rho in studies measuring preferences, attractiveness ratings, or when the underlying scale is not truly interval.
当存在许多相同等级时，需要应用校正因子。心理学家在研究偏好、吸引力评级或当底层量表并非真正的等距时，使用斯皮尔曼 ρ。
7. The t‑test: Independent and Related Samples | t 检验：独立样本与相关样本
The t‑test evaluates whether the means of two groups differ significantly. The formula shifts depending on whether the data come from independent groups or repeated measures / matched pairs.
t 检验用于评估两组的均值是否存在显著差异。根据数据来自独立组还是重复测量/配对组，公式会有所不同。
Independent: t = (X̄₁ – X̄₂) / √[sₚ²(1/n₁ + 1/n₂)]
where the pooled variance sₚ² = [(n₁-1)s₁² + (n₂-1)s₂²] / (n₁+n₂-2). For related designs, the paired t‑test uses the mean and standard deviation of the difference scores (D).
其中合并方差 sₚ² = [(n₁-1)s₁² + (n₂-1)s₂²] / (n₁+n₂-2)。对于相关设计，配对 t 检验使用差值分数 (D) 的均值和标准差。
Related: t = X̄ᴅ / (sᴅ/√n)
The calculated t is compared against critical values from the t‑distribution with df = n₁+n₂-2 (independent) or df = n-1 (related). A significant result (p < .05) suggests that the observed difference is unlikely under the null hypothesis.
计算得到的 t 值与 t 分布的临界值进行比较，独立样本的自由度为 df = n₁+n₂-2，相关样本为 df = n-1。显著结果（p < .05）表明在零假设下观察到的差异不太可能出现。
8. Chi‑Square Test of Association | 卡方检验
The chi‑square (χ²) test for independence determines whether there is a significant association between two categorical variables. It compares observed frequencies with expected frequencies under the assumption of no relationship.
卡方（χ²）独立性检验判断两个分类变量之间是否存在显著关联。它将观察频数与假设无关联时的期望频数进行比较。
χ² = Σ [(O – E)² / E]
Where O is the observed frequency in each cell of a contingency table and E is the expected frequency, calculated as (row total × column total) / grand total. The degrees of freedom are (rows-1)×(columns-1).
其中 O 为列联表每个单元格中的观察频数，E 为期望频数，计算公式为 (行总计 × 列总计) / 总计。自由度为 (行数-1)×(列数-1)。
A significant χ² indicates that the distribution of one variable depends on the other, but it does not show the strength of the association. Effect size measures such as Cramér’s V are often reported alongside χ² to address this limitation.
显著的 χ² 表明一个变量的分布依赖于另一个变量，但它并不显示关联的强度。效应量指标如 Cramér’s V 常与 χ² 一起报告以弥补这一局限。
9. Non‑parametric Alternatives: Mann‑Whitney U and Wilcoxon | 非参数替代检验：Mann‑Whitney U 与 Wilcoxon 符号秩检验
When the assumptions of a t‑test are violated—particularly with skewed data or ordinal measures—researchers turn to rank‑based alternatives. The Mann‑Whitney U test compares two independent groups, while the Wilcoxon signed‑rank test serves related designs.
当 t 检验的假设被违反时——尤其是数据偏态或只有顺序测量——研究者转向基于等级的替代方法。Mann‑Whitney U 检验比较两个独立组，Wilcoxon 符号秩检验则用于相关设计。
U = n₁n₂ + [n₁(n₁+1)/2] – R₁
where R₁ is the sum of ranks for group 1. The smaller of U and U’ (calculated symmetrically) is compared against critical values. This test detects differences in central tendency without assuming normality.
其中 R₁ 为组 1 的秩和。取 U 与 U’（对称计算）中较小的值与临界值比较。该检验无需假设正态性便可检测集中趋势的差异。
The Wilcoxon signed‑rank test computes T = min(T⁺, T⁻), where T⁺ and T⁻ are sums of positive and negative signed ranks of the difference scores. Both tests are common in Pre‑U practical investigations when using rating scales or skewed reaction time data.
Wilcoxon 符号秩检验计算 T = min(T⁺, T⁻)，T⁺ 和 T⁻ 分别为差值分数正、负符号秩之和。当使用评定量表或偏态反应时数据时，这两种检验在 Pre‑U 实践研究中很常见。
10. Effect Size and Confidence Intervals | 效应量与置信区间
Beyond statistical significance, modern reporting standards require effect sizes to quantify the practical importance of results and confidence intervals to estimate population parameters. The most common effect size for comparing two means is Cohen’s d.
在统计显著性之外，现代报告标准要求用效应量来量化结果的实际重要性，并用置信区间估计总体参数。比较两个均值时最常见的效应量是 Cohen’s d。
Cohen’s d = (X̄₁ – X̄₂) / sₚ
where sₚ is the pooled standard deviation. Conventions label d = 0.2 small, 0.5 medium, and 0.8 large. For a 95% confidence interval around a mean, we use X̄ ± (t_crit × SE), where SE = s/√n.
其中 sₚ 为合并标准差。惯例将 d = 0.2 标记为小，0.5 为中，0.8 为大。围绕均值的 95% 置信区间计算公式为 X̄ ± (t_crit × SE)，其中 SE = s/√n。
Confidence intervals convey the precision of an estimate: a narrow CI indicates a reliable estimate, while a wide CI suggests large sampling error. Together, effect sizes and CIs offer a richer picture than p‑values alone, aligning with the Pre‑U emphasis on critical evaluation of data.
置信区间传达估计的精度：窄 CI 表示估计可靠，宽 CI 则提示较大的抽样误差。效应量与置信区间共同提供了比单纯 p 值更丰富的信息，这与 Pre‑U 强调对数据的批判性评估相一致。
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